Reduction of the N-particle variational problem

A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′) = (ψ |α₂+α₁+α _1′α_2′). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N - 1) and trγ = N] then the associated operator: G(1, 2 | 1′, 2′) = δ(1 - 1′)γ(2 | 2′) + σ Γ(1′, 2 | 1, 2′) - γ(2 | 1)7(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or - 1 for bosons or fermions respectively.].